I want to change the frame by doing translation and rotation. $$f(\vec{v})=\sum_{n,l,m}R_{nl}(v)Y_{lm}(\hat{v})f_{nlm}^v.$$ Let, $\mathcal{R}$ be the rotation matrix and $\mathcal{T}$ be the translation operator. I know the rotation acting on $f(\vec{v})$ acts in the following way: $$\mathcal{R}.f(\vec{v})=\sum_{n,l,m}R_{nl}(v)(\sum_{m'}G^{l}_{mm'}Y_{lm'}(\hat{v}))f_{nlm}^v$$ where $G^{l}_{mm'}$ is called the Wigner-$G$ matrix. I do not know what is the analogous expression for the translation operator $\mathcal{T}$ to act on the function $f(\vec{v})$.
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